
(8.3.12) of [1]} proves the moment bound \(| \mathbf {E}\bar{\varepsilon }_k | \le 1/(mv_n)\) . The second moment of \(\bar{\varepsilon }_k\)  is bounded as follows:
\( \phantom{\,.} \mathbf {E}| \bar{\varepsilon }_k|^2 & \lesssim \mathbf {E}| \bar{\varepsilon }_k - \mathbf {E}(\bar{\varepsilon }_k | X_{nk}) |^2 + \mathbf {E}| \mathbf {E}( \bar{\varepsilon }_k | X_{nk} ) - \mathbf {E}\bar{\varepsilon }_k |^2 + | \mathbf {E}\bar{\varepsilon }_k |^2 \, ,\) 
